277:
17:
416:| is summable over the non-identity elements of the group. In fact taking a closed disk in the interior of the fundamental domain, its images under different group elements are disjoint and contained in a fixed disk about 0. So the sums of the areas is finite. By the changes of variables formula, the area is greater than a constant times |
428:
A similar argument implies that the limit set has
Lebesgue measure zero. For it is contained in the complement of union of the images of the fundamental region by group elements with word length bounded by
399:
728:
Vorlesungen über die
Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen.
1092:
433:. This is a finite union of circles so has finite area. That area is bounded above by a constant times the contribution to the Poincaré sum of elements of word length
264:
that all finitely generated classical
Schottky groups have limit sets of Hausdorff dimension bounded above strictly by a universal constant less than 2. Conversely,
1046:
968:
736:
708:
990:
1080:
890:
320:
268:
has proved that there exists a universal lower bound on the
Hausdorff dimension of limit sets of all non-classical Schottky groups.
312:
184:) in the Riemann sphere is given by the exterior of the Jordan curves defining it. The corresponding quotient space Ω(
248:
if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. Marden (
276:
1072:
1130:
845:
757:
162:
953:
Marden, A. (1977), "Geometrically finite
Kleinian groups and their deformation spaces", in Harvey, W. J. (ed.),
1202:
1192:
1067:
170:
112:
581:
771:
700:
Vorlesungen über die
Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen.
473:−3. It contains classical Schottky space as the subset corresponding to classical Schottky groups.
1197:
1172:
901:
621:
824:
481:
256:) gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and
1207:
311:
The statement on
Lebesgue measures follows for classical Schottky groups from the existence of the
769:
Hou, Yong (2010), "Kleinian groups of small
Hausdorff dimension are classical Schottky groups I",
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926:
814:
811:
All finitely generated
Kleinian groups of small Hausdorff dimension are classical Schottky groups
798:
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37:
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is not simply connected in general, but its universal covering space can be identified with
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the "exterior" of the curve, and the other piece its "interior". Suppose there are 2
885:, Mathematical Surveys and Monographs, vol. 8, American Mathematical Society,
619:
Chuckrow, Vicki (1968), "On
Schottky groups with applications to kleinian groups",
549:
Akaza, Tohru (1964), "Poincaré theta series and singular sets of Schottky groups",
77:
61:
1143:
1037:, Grundlehren der Mathematischen Wissenschaften, vol. 287, Berlin, New York:
858:
465:) that generate a Schottky group, up to equivalence under Möbius transformations (
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954:
502:
25:
563:
166:
16:
1151:
1113:
1105:
1013:
922:
899:
Marden, Albert (1974), "The geometry of finitely generated kleinian groups",
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604:
68:
divides the Riemann sphere into two pieces, and we call the piece containing
308:< 2. It is perfect and nowhere dense with positive logarithmic capacity.
285:
794:
169:, has nonempty domain of discontinuity, and all non-trivial elements are
1128:
Yamamoto, Hiro-o (1991), "An example of a nonclassical Schottky group",
1088:"Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen"
956:
Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975)
1004:
930:
681:
650:
200:. This is the boundary of the 3-manifold given by taking the quotient (
161:, a finitely generated Kleinian group is Schottky if and only if it is
785:
914:
634:
192:
is given by joining up the Jordan curves in pairs, so is a compact
819:
275:
15:
579:
Bers, Lipman (1975), "Automorphic forms for Schottky groups",
394:{\displaystyle \displaystyle {P(z)=\sum (c_{i}z+d_{i})^{-4}.}}
665:
Doyle, Peter (1988), "On the bass note of a Schottky group",
111:
in the Riemann sphere with disjoint interiors. If there are
149:
is any Kleinian group that can be constructed like this.
141:, then the group generated by these transformations is a
176:
A fundamental domain for the action of a Schottky group
260:
gave an explicit example of one. It has been shown by
449:≥ 2) is the space of marked Schottky groups of genus
324:
323:
469:). It is a complex manifold of complex dimension 3
393:
232:can be obtained from some Schottky group of genus
228:. Conversely any compact Riemann surface of genus
20:Fundamental domain of a 3-generator Schottky group
1173:Three transformations generating a Schottky group
988:(1967), "A characterization of Schottky groups",
1093:Journal für die reine und angewandte Mathematik
883:Discontinuous Groups and Automorphic Functions
8:
280:Schottky (Kleinian) group limit set in plane
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240:Classical and non-classical Schottky groups
841:"The boundary of classical Schottky space"
1068:Indra's Pearls: The Vision of Felix Klein
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288:of a Schottky group, the complement of Ω(
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731:(in German), Leipzig: B. G. Teubner.,
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453:, in other words the space of sets of
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1065:, Caroline Series, and David Wright,
725:Fricke, Robert; Klein, Felix (1912),
703:(in German), Leipzig: B. G. Teubner,
697:Fricke, Robert; Klein, Felix (1897),
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991:Journal d'Analyse Mathématique
445:Schottky space (of some genus
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1144:10.1215/S0012-7094-91-06308-8
859:10.1215/s0012-7094-79-04619-2
272:Limit sets of Schottky groups
216:space plus the regular set Ω(
596:10.1016/0001-8708(75)90117-6
296:zero, but can have positive
212:of 3-dimensional hyperbolic
835:Jørgensen, T.; Marden, A.;
759:A Survey of Schottky Groups
244:A Schottky group is called
1224:
1073:Cambridge University Press
1131:Duke Mathematical Journal
846:Duke Mathematical Journal
564:10.1017/S0027763000011338
407:showed that the series |
1106:10.1515/crll.1877.83.300
476:Schottky space of genus
220:) by the Schottky group
180:on its regular points Ω(
1177:Fricke & Klein 1897
881:Lehner, Joseph (1964),
772:Geometry & Topology
582:Advances in Mathematics
795:10.2140/gt.2010.14.473
395:
281:
123:taking the outside of
113:Möbius transformations
38:Friedrich Schottky
21:
1086:Schottky, F. (1877),
902:Annals of Mathematics
622:Annals of Mathematics
437:, so decreases to 0.
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32:is a special sort of
19:
963:, pp. 259–293,
321:
64:not passing through
36:, first studied by
829:2013arXiv1307.2677H
132:onto the inside of
1005:10.1007/BF02788719
809:Hou, Yong (2013),
682:10.1007/bf02392277
488:Riemann surfaces.
391:
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163:finitely generated
22:
1048:978-3-540-17746-3
970:978-0-12-329950-5
905:, Second Series,
738:978-1-4297-0552-3
710:978-1-4297-0551-6
625:, Second Series,
498:Beltrami equation
484:of compact genus
482:Teichmüller space
302:Hausdorff measure
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1100:: 300–351,
998:: 227–230,
779:: 473–519,
675:: 249–284,
522:Lehner 1964
503:Riley slice
157:By work of
26:mathematics
1208:Lie groups
1187:Categories
947:0282.30014
747:32.0430.01
719:28.0334.01
543:References
534:Akaza 1964
266:Hou (2010)
171:loxodromic
153:Properties
48:Definition
1152:0012-7094
1122:118718425
1114:0075-4102
1014:0021-7670
923:0003-486X
867:0012-7094
820:1307.2677
803:119144655
643:0003-486X
605:0001-8708
573:118640111
557:: 43–65,
467:Bers 1975
379:−
342:∑
286:limit set
246:classical
196:of genus
76:disjoint
1031:(1988),
839:(1979),
492:See also
405:Poincaré
1160:1106942
1075:, 2002
1057:0959135
1022:0220929
979:0494117
939:0349992
931:1971059
875:0534060
825:Bibcode
691:0945013
659:0227403
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1148:ISSN
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